A block-iterative surrogate constraint splitting method for quadratic signal recovery

IEEE Transactions on Signal Processing

Volumn 51, Issue 7, 2003, Pages 1771-1782

  Combettes, Patrick L ^a  

^a UNIVERSITÉ PIERRE ET MARIE CURIE   (France)

Author keywords

Block iterative optimization; Convex analysis; Deconvolution; Quadratic programming; Signal recovery; Subgradient projection

Indexed keywords

CONSTRAINT THEORY; CONVOLUTION; ITERATIVE METHODS; OPTIMIZATION; PARALLEL PROCESSING SYSTEMS; PROBLEM SOLVING; QUADRATIC PROGRAMMING;

SIGNAL RECOVERY;

SIGNAL RECONSTRUCTION;

EID: 0038538416     PISSN: 1053587X     EISSN: None     Source Type: Journal    
DOI: 10.1109/TSP.2003.812846     Document Type: Review

References (54)

1. E. Artzy, T. Elfving, and G. T. Herman, "Quadratic optimization for image reconstruction II," Comput. Graph. Image Processing, vol. 11, pp. 242-261, 1979.
2. H. H. Bauschke and J. M. Borwein, "Dykstra's alternating projection algorithm for two sets," J. Approx. Theory, vol. 79, pp. 418-443, 1994.
3. _, "On projection algorithms for solving convex feasibility problems," SIAM Rev., vol. 38, pp. 367-426, Sept. 1996.
4. H. H. Bauschke and P. L. Combettes, "A weak-to-strong convergence principle for Fejér-monotone methods in Hubert spaces," Math. Oper. Res., vol. 26, pp. 248-264, May 2001.
5. J. P. Boyle and R. L. Dykstra, "A method for finding projections onto the intersection of convex sets in Hubert spaces," Lecture Notes Statist., vol. 37, pp. 28-47, 1986.
6. J. M. Borwein and A. S. Lewis, Convex Analysis and Nonlinear Optimization - Theory and Examples. New York: Springer-Verlag, 2000.
7. Y. Censor, "Finite series-expansion reconstruction methods," Proc. IEEE, vol. 71, pp. 409-419, Mar. 1983.
8. Y. Censor and S. A. Zenios, Parallel Optimization: Theory, Algorithms and Applications. New York: Oxford Univ. Press, 1997.
9. P. L. Combettes, "Signal recovery by best feasible approximation," IEEE Trans. Image Processing, vol. 2, pp. 269-271, Apr. 1993.
10. _, "Inconsistent signal feasibility problems: Least-squares solutions in a product space," IEEE Trans. Signal Processing, vol. 42, pp. 2955-2966, Nov. 1994.
11. _, "Construction d'un point fixe commun à une famille de contractions fermes," C. R. Acad. Sci. Paris Sér. I Math., vol. 320, pp. 1385-1390, June 1995.
12. _, "The convex feasibility problem in image recovery," in Advances in Imaging and Electron Physics, P. Hawkes, Ed. New York: Academic, 1996, vol. 95, pp. 155-270.
13. _, "Convex set theoretic image recovery by extrapolated iterations of parallel subgradient projections," IEEE Trans. Image Processing, vol. 6, pp. 493-506, Apr. 1997.
14. _, "Strong convergence of block-iterative outer approximation methods for convex optimization," SIAM J. Contr. Optim., vol. 38, pp. 538-565, Feb. 2000.
15. P. L. Combettes and P. Bondon, "Hard-constrained inconsistent signal feasibility problems," IEEE Trans. Signal Processing, vol. 47, pp. 2460-2468, Sept. 1999.
16. P. L. Combettes and T. J. Chaussalet, "Combining statistical information in set theoretic estimation," IEEE Signal Processing Lett., vol. 3, pp. 61-62, Mar. 1996.
17. P. L. Combettes and J. Luo, "An adaptive level set method for nondifferentiable constrained image recovery," IEEE Trans. Image Processing, vol. 11, pp. 1295-1304, Nov. 2002.
18. P. L. Combettes and H. J. Trussell, "The use of noise properties in set theoretic estimation," IEEE Trans. Signal Processing, vol. 39, pp. 1630-1641, July 1991.
19. F. Deutsch, Best Approximation in Inner Product Spaces. New York: Springer-Verlag, 2001.
20. S. Dharanipragada and K. S. Arun, "A quadratically convergent algorithm for convex set constrained signal recovery," IEEE Trans. Signal Processing, vol. 44, pp. 248-266, Feb. 1996.
21. R. L. Dykstra, "An algorithm for restricted least squares regression," J. Amer. Statist. Assoc., vol. 78, pp. 837-842, Dec. 1983.
22. B. Eicke, "Iteration methods for convexly constrained ill-posed problems in Hubert space," Numer. Func. Anal. Optim., vol. 13, pp. 413-429, 1992.
23. M. Elad and A. Feuer, "Restoration of a single superresolution image from several blurred, noisy, and undersampled measured images," IEEE Trans. Image Processing, vol. 6. pp. 1646-1658, Dec. 1997.
24. N. Gaffke and R. Mathar, "A cyclic projection algorithm via duality," Metrika, vol. 36, pp. 29-54, 1989.
25. U. M. García-Palomares and F. J. González-Castaño, "Incomplete projection algorithms for solving the convex feasibility problem," Numer. Algorithms, vol. 18, pp. 177-193, 1998.
26. F. J. González-Castafio, U. M. García-Palomares, J. L. Alba-Castro, and J. M. Pousada-Carballo, "Fast image recovery using dynamic load balancing in parallel architectures, by means of incomplete projections," IEEE Trans. Image Processing, vol. 10, pp. 493-499, Apr. 2001.
27. B. Halpern, "Fixed points of nonexpanding maps," Bull. Amer. Math. Soc., vol. 73, pp. 957-961, Nov. 1967.
28. Y. Haugazeau, "Sur les Inéquations Variatipnnelles et la Minimisation de Fonctionnelles Convexes," Ph.D. dissertation, Univ. Paris, Paris, France, 1968.
29. G. T. Herman and A. Lent, "Quadratic optimization for image reconstruction I," Comput. Graph. Image Processing, vol. 5, pp. 319-332, 1976.
30. B. R. Hunt, "The inverse problem of radiography," Math. Biosci.,vol 8, pp. 161-179, June 1970.
31. _, "The application of constrained least-squares estimation to image restoration by digital computer," IEEE Trans. Comput., vol. C-22, pp. 805-812, Sept. 1973.
32. A. N. Iusem and A. R. De Pierro, "On the convergence of Han's method for convex programming with quadratic objective," Math. Programming, vol. 52, pp. 265-284, Aug. 1991.
33. A. N. Iusem and B. F. Svaiter, "A row-action method for convex programming," Math. Programming, vol. 64, pp. 149-171, 1994.
34. A. K. Katsaggelos, "A multiple input image restoration approach," J. Visual Commun. Image Represent, vol. 1, pp. 93-103, Sept. 1990.
35. K. C. Kiwiel, "Monotone Gram matrices and deepest surrogate inequalities in accelerated relaxation methods for convex feasibility problems," Linear Algebra Appl, vol. 252, pp. 27-33, Feb. 1997.
36. H. Kudo and T. Saito, "Sinogram recovery with the method of convex projections for limited-data reconstruction in computed tomography," J. Opt. Soc. Amer.,vol. A8, pp. 1148-1160, July 1991.
37. M. Lapidus, "Généralisation de la formule de Trotter-Lie," C. R. Acad. Sci. Paris Sér. A Math., vol. 291, pp. 497-500, Oct. 1980.
38. R. M. Leahy and C. E. Goutis. "An optimal technique for constraint-based image restoration and reconstruction," IEEE Trans. Acoust., Speech, Signal Processing, vol. ASSP-34, pp. 1629-1642, Dec. 1986.
39. A. J. Levy, "A fast quadratic programming algorithm for positive signal restoration," IEEE Trans. Acoust., Speech, Signal Processing, vol. ASSP-31, pp. 1337-1341, Dec. 1983.
40. M. K. Ng and W. C. Kwan, "Comments on "Least squares restoration of multichannel images"," IEEE Trans. Signal Processing, vol. 49, p. 2885, Nov. 2001.
41. P. Oskoui-Fard and H. Stark, "Tomographie image reconstruction using the theory of convex projections," IEEE Trans. Med. Imag., vol. 7, pp. 45-58, Mar. 1988.
42. G. Pierra, "Éclatement de contraintes en parallèle pour la minimisation d'une forme quadratique," in Lecture Notes in Comput. Sci., 1976, vol. 41, pp. 200-218. See also "Decomposition through formalization in a product space," Math. Programming, vol. 28, pp. 96-115, Jan. 1984.
43. G. Pierra, "Éclatement de contraintes en parallèle pour la minimisation d'une forme quadratique," in Lecture Notes in Comput. Sci., 1976, vol. 41, pp. 200-218. See also "Decomposition through formalization in a product space," Math. Programming, vol. 28, pp. 96-115, Jan. 1984.
44. S. Reich, "A limit theorem for projections," Linear Multilinear Algebra, vol. 13, pp. 281-290, 1983.
45. A. Sabharwal and L. C. Potter, "Convexly constrained linear inverse problems: Iterative least-squares and regularization," IEEE Trans. Signal Processing, vol. 46, pp. 2345-2352, Sept. 1998.
46. L. Schwartz, Analyse - Topologie GÉnérale et Analyse Fonctionnelle, 2nd ed. Paris, France: Herman, 1980.
47. M. I. Sezan and H. J. Trussell, "Prototype image constraints for settheoretic image restoration," IEEE Trans. Signal Processing, vol. 39, pp. 2275-2285, Oct. 1991.
48. D. M. Titterington, "General structure of regularization procedures in image reconstruction," Astron. Astrophys., vol. 144, pp. 381-387, Mar. 1985.
49. H. J. Trussell and M. R. Civanlar, "The feasible solution in signal restoration," IEEE Trans. Acoust., Speech, Signal Processing, vol. ASSP-32, pp. 201-212, Apr. 1984.
50. S. Twomey, "The application of numerical filtering to the solution of integral equations encountered in indirect sensing measurements," J. Franklin Inst., vol. 279, pp. 95-109, Feb. 1965.
51. I. Yamada, N. Ogura, Y. Yamashita, and K. Sakaniwa, "Quadratic optimization of fixed points of nonexpansive mappings in Hubert space," Numer. Funct. Anal. Optim., vol. 19, pp. 165-190, 1998.
52. I. Yamada, K. Slavakis, and K. Yamada, "An efficient robust adaptive filtering algorithm based on parallel subgradient projection techniques," IEEE Trans. Signal Processing, vol. 50, pp. 1091-1101, May 2002.
53. D. C. Youla and H. Webb, "Image restoration by the method of convex projections: Part 1 - Theory," IEEE Trans. Medical Imag., vol. MI-1, pp. 81-94, Oct. 1982.
54. E. Zeidler, Nonlinear Functional Analysis and Its Applications III: Variational Methods and Optimization. New York: Springer-Verlag, 1985.